Problem 22 - College Algebra

Question

For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points. $$(-5,-6)\text { and }(4,2)$$

Step-by-Step Solution

Answer

The midpoint is \((-0.5, -2)\).

Step 1: Identify the Given Points

We are provided with two points: \((-5, -6)\) and \((4, 2)\). These points represent the endpoints of the line segment.

Step 2: Use the Midpoint Formula

The formula for finding the midpoint \((x, y)\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\).

Step 3: Calculate the x-coordinate of the Midpoint

Using the x-coordinates of the given points, calculate the average: \(\frac{-5 + 4}{2} = \frac{-1}{2} = -0.5\).

Step 4: Calculate the y-coordinate of the Midpoint

Using the y-coordinates of the given points, calculate the average: \(\frac{-6 + 2}{2} = \frac{-4}{2} = -2\).

Step 5: Combine the Midpoint Coordinates

The calculated midpoint coordinates are \((-0.5, -2)\). This represents the midpoint of the line segment joining the two given points.

Key Concepts

Coordinate Geometry

Coordinate geometry is an essential branch of mathematics that allows you to study the geometric figures using a coordinate system. It's a way to visualize and calculate the properties of geometric shapes using algebraic techniques.
In coordinate geometry, points are described using pairs of numbers in the form \(x, y\). This helps you locate any point on a plane uniquely.
Using coordinate geometry, you can determine distances, angles, and other properties of geometric figures. It forms the basis for understanding various geometric concepts and solving real-world problems using algebraic methods.

Line Segment

A line segment is a part of a line that is bounded by two endpoints. Unlike a line, it does not extend infinitely. In the context of coordinate geometry, a line segment can be represented by its endpoints' coordinates.
For example, given two points \(A(-5, -6)\) and \(B(4, 2)\), the line segment is the shortest path connecting these points. It is often necessary to find the midpoint or length of a line segment in various mathematical problems.

• Endpoints: Points that define the beginning and end of the segment.
• Midpoint: The centermost point of a line segment, equidistant from both endpoints.

These concepts make it easier to work with sections of lines in mathematical calculations.

Calculating Midpoints

Calculating midpoints is a standard problem in coordinate geometry that involves finding the middle point of a line segment. The midpoint formula is a critical tool for finding the center point between two given endpoints.
To find the midpoint of a line segment with endpoints \(x_1, y_1\) and \(x_2, y_2\), you use the formula: \[\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\].
This formula simply calculates the average of the x-coordinates and the y-coordinates separately.

• Example: For points \((-5, -6)\) and \((4, 2)\), the midpoint is found as follows:
  • Calculate the x-coordinate: \(\frac{-5 + 4}{2} = -0.5\)
  • Calculate the y-coordinate: \(\frac{-6 + 2}{2} = -2\)
• The midpoint is \((-0.5, -2)\).

This midpoint is directly in between the two given endpoints.

Coordinate System

A coordinate system is a framework to uniquely determine each point in a plane through numerical coordinates. The most commonly used system in geometry is the Cartesian coordinate system, which is defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
The intersection of these axes is known as the origin, denoted as \(0, 0\). Each point on the plane can be located using an ordered pair \(x, y\). The x-coordinate indicates the horizontal distance from the origin, and the y-coordinate indicates the vertical distance.

• X-axis: Horizontal line where y = 0
• Y-axis: Vertical line where x = 0
• Quadrants: The plane is divided into four quadrants by the axes.
• Positive and Negative Values: Depending on the position, values can be positive or negative.

Understanding the coordinate system is crucial for accurately finding the location of points and solving geometric problems involving distance, slopes, and midpoints.